Global normally hyperbolic invariant cylinders in Lagrangian systems

Abstract

In this paper, we study Tonelli Lagrangian L ∈ C r (T T 2 , R) with r ≥ 5. For a generic perturbation of Lagrangian L → L + P where P ∈ C r (T 2 , R), we get simultaneous hyperbolicity of a family of minimal periodic orbits which share the same first homology class. Consequently, these periodic orbits make up one or more pieces of normally hyperbolic invariant cylinder in T T 2 .

Uniform hyperbolicity of invariant cylinder

“…The result in [CZ1] plays important role in this paper. It is for the minimal periodic orbit of Tonelli Lagrangian of two degrees of freedom.…”

“…As it was shown in [CZ1], the hyperbolicity of such minimal periodic orbit is uniquely determined by the nondegeneracy of the minimal point of the following function…”

“…As the first step, let us consider the Hamiltonian G i . Applying Theorem 1.3 proved in [CZ1], we find that all (E, g)-minimal periodic orbits make up some pieces of normally hyperbolic invariant cylinders. However, it is not enough to study the persistence of these NHICs under the small perturbation…”

“…The following result has been proved (Theorem 2.1 of [CZ1]): Theorem 1.3. Given a class g ∈ H 1 (T 2 , Z) and two positive numbers E ′′ > E ′ > 0, there exists an open-dense set V ⊂ C r (T 2 , R) with r ≥ 5 such that for each V ∈ V, it holds simultaneously for all E ∈ [E ′ , E ′′ ] that every (E, g)-minimal periodic orbit of L + V is hyperbolic.…”

“…) reaches its minimum at x * , then dγ(·, E, g, x * ) is the minimal periodic orbit we are looking for [CZ1].…”